I include Warm ups with a Rubric as part of my daily routine. My goal is to allow students to work on Math Practice 3 each day. Grouping students into homogeneous pairs provides an opportunity for appropriately differentiated math conversations. The Video Narrative explains this lesson’s Warm Up-Modeling Average Temperature which asks students to analyze the statements of three students who are discussing modeling situations with sine or cosine.
I also use this time to correct and record the previous day's Homework.
This activity takes the average temperature from the town in which my students live and asks them to write a trigonometric equation to model it. This activity can easily be altered to fit your area. I obtained my information from this website for the Western United States. You can get information for places around the world here. Each student will need graphing technology and a piece of graph paper for this lesson.
After an introduction on average temperature, I ask my students to draw a scatter plot of the average overall temperatures in Caldwell, Idaho for two years. Each student receives a slip of paper with the table of values. I contemplated having the students make only one for every pair but I think this is a valuable enough activity for all students to do themselves. The graphs will be used on the homework as well. These temperatures are the monthly averages for the daily highs from 1904 to 1996. It is important for the students to know that this is an OVERALL average for many years and not just one year. I tell them to label the x-axis as months with January on the 1 unit and that the domain should be [0,24]. The range is labeled temperature in degrees Celsius with a range of [0,100].
Once they have drawn the graph, I ask them to discuss the shape with their partner. The goal is that they recognize the fact that this is a cosine function. Some may say sine which is correct however they should recognize this as cosine as well since it was just used in the Ferris Wheel activity (Math Practice 5). We then discuss as a class. Both the partner sharing and the class discussion should be quick. Finally, they draw a curve through the points.
We are going to use following model to write a trig function to model this graph: What do a, b, c and d do to a cosine function? I have the students talk in pairs to answer this question (Math Practice 4). My kids have seen this a bunch so this should only take a minute or two. We then go over it as a class. I call on people randomly to describe each transformation.
Next, I give the students some time to figure out the four values. Some scaffolding that could be done at this point would be to draw a simple model on the board with some transformations and then show them how to find the equation. I stop at points and give hints to the entire class if many pairs seem to be struggling. For example, if the students are having a hard time with the period (c) then I remind them that the usual period is 360 degrees. I ask them what c would have to be to make the period go to 180 degrees (the correct answer would be two). One way to look at this would be to say how many 180s make a 360. If our period is 12 units then how many of those will fill up 360? (Math Practice 7)
It is important not to give answers but only hints at this point. I encourage them and ask guiding questions like the one above.
This next section is done on a TI 84 calculator. The students are making a table and graphing it as a scatter plot. I chose not to print out the calculator instructions to give to students but feel free make a handout for your students. The goal of using the calculator is to obtain a check of our equation model and alter it if necessary. It won’t fit exactly (this will be dealt with later) but it should be as close as students feel comfortable.
These instructions tell the student how to set up and create a scatter plot and equation model on their calculator:
I will usually model this on my overhead. This activity is a great way to train students in Math Practice 5. I recommend plotting your data by hand and by calculator to both prepare a model and get yourself ready to head off any problems students may come across.
Once they have a model in place they can shift it until they get something they feel fits the data as closely as possible. This should be discussed with their partner although both should be inputting numbers. Finally each pair writes their model on the white board. If the pair can't agree then each should write their own.
Once all of the models have been put on the board, we discuss them as a class, noting trends. This portion will all depend on the models that students wrote. They should be fairly similar but there will probably be some differences. This is an opportunity for students to defend their position as well as critique the reasoning of others (Math Practice 3). I am always enthusiastic about differences, even when they are incorrect, and try to encourage students to look at their mistakes as learning experiences. When dealing with a glaring error, it is good to talk about it but depending on the students, it may be wise to discuss it without calling them out. A possible way to deal with an error would be to ask students to consider why this mistake was made (again enthusiasm goes a long way here to dispel embarrassment).
Another important conversation to have would be about which model is the “best”. While some issues will be obvious (like period), others will not be quite as clear. The vertical transformation may be slightly different. The same for amplitude. It is important for students to realize that there isn’t always a single right answer (Math Practice 5). Particularly with modeling, there may be many good solutions.
One other thing to discuss would be why the shape didn’t match exactly. The actual answer lies in the fact that the earth’s orbit is elliptical therefore our seasons don’t lie on a cosine since cosine models circular behavior.
This Homework wraps up the modeling activity for the students. The first several questions ask evaluative questions about the graph and the model. These questions provide opportunities to use Math Practice 2 and Math Practice 7. There is also a question that asks the students to graph the average monthly temperature for a single year. The students are then asked to describe the weather that year in comparison to the overall average weather. The final question gives the students a graph of the average temperatures in Chicago and San Diego. The students are asked to write a paragraph describing what they can determine from this graph.
I use an exit ticket each day as a quick formative assessment to judge the success of the lesson.
This Exit Ticket allows the students to reflect on the parts of the model that were the easiest or most obvious and which provided the most struggle.